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# amat2_psc_exp – Non-resetting leaky integrate-and-fire neuron model with exponential PSCs and adaptive threshold¶

## Description¶

amat2_psc_exp is an implementation of a leaky integrate-and-fire model with exponential shaped postsynaptic currents (PSCs). Thus, postsynaptic currents have an infinitely short rise time.

The threshold is lifted when the neuron is fired and then decreases in a fixed time scale toward a fixed level 3.

The threshold crossing is followed by a total refractory period during which the neuron is not allowed to fire, even if the membrane potential exceeds the threshold. The membrane potential is NOT reset, but continuously integrated.

The linear subthreshold dynamics is integrated by the Exact Integration scheme 1. The neuron dynamics is solved on the time grid given by the computation step size. Incoming as well as emitted spikes are forced to that grid.

An additional state variable and the corresponding differential equation represents a piecewise constant external current.

The general framework for the consistent formulation of systems with neuron like dynamics interacting by point events is described in 1. A flow chart can be found in 2.

Remarks:

• The default parameter values for this model are different from the corresponding parameter values for mat2_psc_exp.

• If identical parameters are used, and beta==0, then this model shall behave exactly as mat2_psc_exp.

• The time constants in the model must fullfill the following conditions: - $$\tau_m != {\tau_{syn_{ex}}, \tau_{syn_{in}}}$$ - $$\tau_v != {\tau_{syn_{ex}}, \tau_{syn_{in}}}$$ - $$\tau_m != \tau_v$$ This is required to avoid singularities in the numerics. This is a problem of implementation only, not a principal problem of the model.

• Expect unstable numerics if time constants that are required to be different are very close.

• $$\tau_m != \tau_{syn_{ex,in}}$$ is required by the current implementation to avoid a degenerate case of the ODE describing the model 1. For very similar values, numerics will be unstable.

## Parameters¶

The following parameters can be set in the status dictionary:

 C_m pF Capacity of the membrane E_L mV Resting potential tau_m ms Membrane time constant tau_syn_ex ms Time constant of postsynaptic excitatory currents tau_syn_in ms Time constant of postsynaptic inhibitory currents t_ref ms Duration of absolute refractory period (no spiking) V_m mV Membrane potential I_e pA Constant input current t_spike ms Point in time of last spike tau_1 ms Short time constant of adaptive threshold [3, eqs 2-3] tau_2 ms Long time constant of adaptive threshold [3, eqs 2-3] alpha_1 mV Amplitude of short time threshold adaption [3, eqs 2-3] alpha_2 mV Amplitude of long time threshold adaption [3, eqs 2-3] tau_v ms Time constant of kernel for voltage-dependent threshold component [3, eqs 16-17] beta 1/ms Scaling coefficient for voltage-dependent threshold component [3, eqs 16-17] omega mV Resting spike threshold (absolute value, not relative to E_L as in 3)
 State variables that can be read out with the multimeter device V_m mV Non-resetting membrane potential V_th mV Two-timescale adaptive threshold

## References¶

1(1,2,3)

Rotter S, Diesmann M (1999). Exact simulation of time-invariant linear systems with applications to neuronal modeling. Biologial Cybernetics 81:381-402. DOI: https://doi.org/10.1007/s004220050570

2

Diesmann M, Gewaltig M-O, Rotter S, & Aertsen A (2001). State space analysis of synchronous spiking in cortical neural networks. Neurocomputing 38-40:565-571. DOI: https://doi.org/10.1016/S0925-2312(01)00409-X

3(1,2)

Kobayashi R, Tsubo Y and Shinomoto S (2009). Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold. Frontiers in Computational Neuroscience, 3:9. DOI: https://dx.doi.org/10.3389%2Fneuro.10.009.2009

4

Yamauchi S, Kim H, Shinomoto S (2011). Elemental spiking neuron model for reproducing diverse firing patterns and predicting precise firing times. Frontiers in Computational Neuroscience, 5:42. DOI: https://doi.org/10.3389/fncom.2011.00042

SpikeEvent